Evaluation of $$\int^{\frac{\pi}{2}}_{0}\frac{1}{\sqrt{1+\sqrt{\tan x}}}dx$$
Try: Put $\tan x=t^2$ and $\sec^2 x dx=2tdt$.
$$I=\int^{\infty}_{0}\frac{2t}{(1+t^4)\sqrt{1+t}}dt$$
Put $1+t=u^2$ and $dt=2udu$
$$I=\int^{\infty}_{1}\frac{4(u^2-1)}{1+(u^2-1)^4}du$$
Could some help me to solve it , Thanks